Measures of Polarisation

In broad terms, polarisation has been measured in two kinds of ways – one reflecting the idea of

“ideological variance” or standard deviation already present in the work of Downs (1957, p. 100) and the other reflecting a very close yet distinct notion of an aggregate amount of distance between political parties when compared to each other in pairs, passingly mentioned also by Sartori (2005, p. 106). Neither of the classics gave an actual form, no matter how obvious, to a measure of polarisation, but the research that was building on their ideas and concepts elaborated on these problems extensively. What follows here is but a brief discussion of that, ignoring the source of information about party locations (like mass or expert surveys or party manifestos), the specific topic of the analysis and role of the polarisation variable (explanandum or explanans), focussing just on the mechanics of transforming party locations or differences into a measure of polarisation. These two kinds of measures mentioned here encompass all parties in the system. Additionally, there are measures, which focus only on some parties, which will be mentioned last.

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Measures of ideological variance / standard deviation

From a statistical point of view it is perhaps easiest to think of polarisation as based on ideological variance, because this is the most basic way that we think about the spread of values (in this case the locations of parties on an ideological dimension) of a variable. If we have a unidimensional and continuous concept of ideology, we thus do not need to think of a fundamentally new measure, we can just borrow one that is already known and used in statistics. The formal definition of variance is the average squared deviation from the mean of the variable and a standard deviation is the square root of that. The idea of measuring polarisation on the basis of the squared distance from the mean of the ideological dimension, either as standard deviation or as variance, has been extensively applied in research on polarisation. In one of the first empirical definitions involving party system polarisation, Taylor and Herman (1971, p. 34) effectively define polarisation on the basis of variance as

Pvar = Pn

i fi(xi −¯x)²

n (5.1)

where n refers to the number of seats in parliament, f_i to the number of seats of partyi,x_i is the ideological position of party i and ¯x is the position of the mean of the seat distribution (they use ordinal positions, i.e. a ranking of the parties in the left-right dimension).

One notable difference from the definition of variance is that here we are taking relative party sizes, as measured by their seat distribution in parliament, into account. Instead of average square distance from the mean we are measuring weighted average square distance, reflecting the idea that parties of varying sizes matter differently. Essentially the same measure has been adopted subsequently by several other authors (e.g. Sigelman and Yough 1978; Lachat 2008; Lupu 2015).

There are some authors who have, instead of ideological variance, opted for a measure of ide-ological standard deviation, which essentially constitutes a square root of the ideide-ological variance measure. The most known application of this has been perhaps by Dalton (2008), but there are others before (e.g. Warwick 1994) and after (e.g. Curini and Hino 2012; Dejaeghere and Dasson-neville 2015; Matakos, Troumpounis, and Xefteris 2015; Han 2015; Singer 2016) who have used this measure or something very similar (Ensley 2012). Polarisation as weighted standard deviation can thus be empirically defined as:

Psd = v u u t

n

X

i

wi(xi−¯x)² (5.2)

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where w_i is the weight of partyi and ¯x is the weighted average of the ideological dimension.

Of course there is no fundamental reason why in this case we should be looking at squared distances from the centre of ideological gravity. For example, van der Eijk, Schmitt, and Binder (2005) use a measure based on absolute distances. If we take the logic of the variance/standard deviation measure apart, then one could have even more slightly different measures, e.g. those based on not the mean on the ideological dimension by the median.

Even though such measures borrow concepts that are familiar for us from statistical analyses, it should also be noted that in one respect (in addition to aggregating the distances in a set of parties) they are reflecting the initial idea of Sartori rather well. They measure distances from the centre of gravity of the distribution. This reflects the idea the polarisation is a configuration were the centre of the political spectrum is opposed on both sides by anti-system parties. This compatibility exists, however, only if there actually is such a centre, because the mean of a distribution can also exist where there is no actual party.

Measures of pairwise ideological distance

The second type of measure that has set the tone for polarisation research takes into account the locations of all parties in the system and nothing else. In comparison to the measures above this so to say cuts out the middle man and goes straight to the source. The standard deviation and variance based measures first have to determine the ideological centre of gravity of a political landscape and will then use the locations of parties in relation to that constructed centre to assess polarisation. It is as if we create an imaginary location on the political landscape, a hypothetical party to compare all other parties to. It might correspond to an actual party, but not necessarily. In contrast to this, there is a class of measures that derive an estimate of polarisation directly from actual party positions on an ideological dimension by comparing parties to each other.

This kind of a measure of party polarisation was first proposed by Gross and Sigelman (1984), who consider all pairwise distances in a party system in the following way (notation has been changed to be compatible with other formulations in this chapter):

Pgs =

n

X

i n

X

j

wj

1−w_i|x_i −xj| (5.3)

wherew_i andw_j are the seat shares of parties i andj andx refers to the position of parties i andj on the ideological dimension in question.

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A more elaborated pairwise distance measure was adopted a decade later from the work of Esteban and Ray (1994), which has gained notably more attention and application. They suggest an index of polarisation, that can be expressed as follows:

Per =K

n

X

i n

X

j

π_i^1+απ_j|y_i−y_j| (5.4)

where K is a positive constant,π_i and π_j are weights, α is a parameter that controls the extent to which weight differences are taken into account, and y_i and y_j are locations on a dimension. This measure can be simplified and generalised

Per =K

n

X

i n

X

j

π_i^1+απ_jd_ij (5.5)

so thatd_ij refers to the distance between two parties on not just one dimension, but in any imaginable political space. The equation can further be simplified, by dropping the K and theα, which simply means that the differences in the sizes of the parties are not additionally taken into account (beyond what is reflected in party sizes used as weights) and that the eventual scale is not further adjusted.

This equation would then have the following form:

Per =

n

X

i n

X

j

wiwjdij (5.6)

where w_i andw_j simply refer to the seat or vote shares that are used as weights for the parties.

The Esteban and Ray measure has found applications in a number of later studies on party system polarisation or political difference between parties, like Rehm and Reilly (2010), Indridason (2011), Han (2015) and many others. However, it also seems that the idea of aggregating pairwise distances, even though it has caught the attention of many party researchers, has not reached an agreement over a specific form. Thus, for example Klingemann (2005) uses a different solution for aggregating pairwise differences, which is, among other things, not weighted by relative party size and Lupu (2015) uses a formulation which weights each pairwise distance by the sum of the relative sizes (vote shares in that case) in the pair divided by the number of parties minus 1.

In addition to being simpler – by not involving the additional step of calculating the centre of gravity of the political space, there is one other clear advantage for the formulations of party system polarisation that takes pairwise distances as input. They are compatible with all possible political spaces, as long as there is an estimate for the distances between the pairs of parties in that space.

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The mean-square-based measures, in contrast, are designed with only one dimension in mind and are not able to provide an estimate of political divergences across dimensions, although it is possible to aggregate the separate polarisations for each dimension.

Other measures of polarisation

In addition to these two most widely used measures – the pairwise measure and the standard deviation/variation-based measure – there are several other operationalisations of party system po-larisation that have been suggested and used in party system research, reflecting other nuances of the concept. One version is the maximum distance among parties in a system, referring to the idea that polarisation is defined by the existence of extreme, anti-system parties on both sides of the ideological spectrum. This measure is used, for example, by Dejaeghere and Dassonneville (2015), Matakos, Troumpounis, and Xefteris (2015) and Andrews and Money (2009). Another measure, which reflects a similar idea, is the proportion of extremist parties in the party system. This has been used among others by King et al. (1990) and Warwick (1994). A tangent measure, which has found application in closely related research into voting behaviour, is the measure of party system compactness suggested by Alvarez and Nagler (2004) that takes also the dispersion of voters into account (and is thus suitable for only such cases, where comparable information about the political positions of voters is available, severely restricting the range of application). This measure has been used in party system polarisation research among others by Ezrow (2008) and Dow (2011).